Distribution functions, extremal limits and optimal transport

TL;DR

This paper explores the extremal problem of maximizing integrals over copulas with uniform marginals, connecting it to optimal transport, combinatorial optimization, and applications in mathematical finance.

Contribution

It links extremal limits of sums with uniform sequences to optimal transport and copula theory, providing new proofs and insights, and discusses applications in finance.

Findings

Characterization of maximizing copulas

Alternative proofs for uniform distribution results

Applications to financial modeling

Abstract

Encouraged by the study of extremal limits for sums of the form $$\lim_{N\to\infty}\frac{1 }{N}\sum_{n=1}^N c(x_n,y_n)$$ with uniformly distributed sequences $\{x_n\},\,\{y_n\}$ the following extremal problem is of interest $$\max_{\gamma}\int_{[0,1]^2}c(x,y)\gamma(dx,dy),$$ for probability measures $\gamma$ on the unit square with uniform marginals, i.e., measures whose distribution function is a copula. The aim of this article is to relate this problem to combinatorial optimization and to the theory of optimal transport. Using different characterizations of maximizing $\gamma$'s one can give alternative proofs of some results from the field of uniform distribution theory and beyond that treat additional questions. Finally, some applications to mathematical finance are addressed.